MinMax Arena
中/EN
P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 11

Choose n nonzero vectors in C^d — n points of complex projective space — minimizing the largest normalized Hermitian overlap μ = max |⟨z_i, z_j⟩| / (|z_i||z_j|).

Instanced = 4, n = 11
ObjectiveMinimize the largest coherence μ
Best known, unproven0.425147265the Game of Sloanes leader board (the table of putatively optimal packings maintained by Jasper, King and Mixon)

Formal definition

  • ContainerComplex d-space C^d; an answer is n points of complex projective space CP^{d-1}
  • SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
  • ObjectiveMinimize μ² = max |⟨z_i,z_j⟩|²/(|z_i|²|z_j|²); moduli and norms squared are rational, compared exactly by cross-multiplication
  • ScoringThe record is ceil(μ²·10¹⁸); the page shows μ, rounded up at the ninth decimal
Challenge this record ↗
VERIFIED CONSTRUCTIONpairwise overlap heat of 11 directions; brighter is closer

Getting a feel for it

Where it is used

Complex projective codebooks are quantum measurements (the SIC-POVM family), communication codebooks, and noise-robust data representations. The further apart the codewords, the better they survive noise and erasures.

Where the frontier is

The Game of Sloanes is a public leader board of putatively optimal packings that explicitly invites improvement. Every sub-problem here is chosen from its still-open rows — the ones with a real gap between the best known value and the lower bound.

Source
Current leader

0.99571906

the largest coherence μ

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

Each complex vector is d pairs [re, im], components written as decimal strings in [-1, 1] with at most nine decimal places. Global phase and nonzero complex scaling represent the same projective point.

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.090909091",
        "0"
      ],
      [
        "0.008264463",
        "0"
      ],
      [
        "0.000751315",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.181818182",
        "0"
      ],
      [
        "0.033057851",
        "0"
      ],
      [
        "0.006010518",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.272727273",
        "0"
      ],
      [
        "0.074380165",
        "0"
      ],
      [
        "0.020285500",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.363636364",
        "0"
      ],
      [
        "0.132231405",
        "0"
      ],
      [
        "0.048084147",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.454545455",
        "0"
      ],
      [
        "0.206611570",
        "0"
      ],
      [
        "0.093914350",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.545454545",
        "0"
      ],
      [
        "0.297520661",
        "0"
      ],
      [
        "0.162283997",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.636363636",
        "0"
      ],
      [
        "0.404958678",
        "0"
      ],
      [
        "0.257700977",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.727272727",
        "0"
      ],
      [
        "0.528925620",
        "0"
      ],
      [
        "0.384673178",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.818181818",
        "0"
      ],
      [
        "0.669421488",
        "0"
      ],
      [
        "0.547708490",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.909090909",
        "0"
      ],
      [
        "0.826446281",
        "0"
      ],
      [
        "0.751314801",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.090909091",
        "0"
      ],
      [
        "0.008264463",
        "0"
      ],
      [
        "0.000751315",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.181818182",
        "0"
      ],
      [
        "0.033057851",
        "0"
      ],
      [
        "0.006010518",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.272727273",
        "0"
      ],
      [
        "0.074380165",
        "0"
      ],
      [
        "0.020285500",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.363636364",
        "0"
      ],
      [
        "0.132231405",
        "0"
      ],
      [
        "0.048084147",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.454545455",
        "0"
      ],
      [
        "0.206611570",
        "0"
      ],
      [
        "0.093914350",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.545454545",
        "0"
      ],
      [
        "0.297520661",
        "0"
      ],
      [
        "0.162283997",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.636363636",
        "0"
      ],
      [
        "0.404958678",
        "0"
      ],
      [
        "0.257700977",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.727272727",
        "0"
      ],
      [
        "0.528925620",
        "0"
      ],
      [
        "0.384673178",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.818181818",
        "0"
      ],
      [
        "0.669421488",
        "0"
      ],
      [
        "0.547708490",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.909090909",
        "0"
      ],
      [
        "0.826446281",
        "0"
      ],
      [
        "0.751314801",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer. · Verifier v1.0.0

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.

No threads yet. Be the first to talk about this problem?